Abstract

The theory of pseudolattices was initiated by Kuznetsov, building on work of many other authors. The canonical example is the numerical Grothendieck group associated to the bounded derived category of coherent sheaves on a smooth variety. In this talk I will introduce a special class of pseudolattices, which may be thought of as numerical Grothendieck groups associated to bounded derived categories of coherent sheaves on smooth rational surfaces which admit a smooth anticanonical divisor. Such pseudolattices may be classified and, perhaps unsurprisingly, their classification parallels the well-known classification of del Pezzo surfaces.

However, this special class of pseudolattices also arises naturally in a second context, associated to a certain class of elliptic Lefschetz fibrations over complex discs. I will show that the classification result above allows one to classify such elliptic Lefschetz fibrations up to symplectomorphism. This setting should be thought of as mirror, in a homological sense, to the original context of smooth rational surfaces admitting smooth anticanonical divisors. This work is joint with Andrew Harder.